Britain's foreign minister Jeremy Hunt has called on his rival Boris Johnson not to be a "coward" by avoiding a live TV debate with him this week.
Stepping up pressure on his rival, Mr Hunt insisted that while he has no interest in discussing Mr Johnson's private life, he wants to challenge him on television over his commitment to taking the UK out of the EU by Halloween.
In an article in the UK Times, Mr Hunt said: "A new prime minister needs the legitimacy of having made his arguments publicly and having them subjected to scrutiny.
"Only then can you walk through the front door of No 10 with your head held high instead of slinking through the back door, which is what Boris appears to want."
"Don't be a coward Boris, man up and show the nation you can cope with the intense scrutiny the most difficult job in the country will involve."
On Sunday, Mr Hunt said that he thought any candidate for prime minister needs to show he can answer "difficult questions".
He said: "What Boris needs to do is to engage properly in this leadership debate, not to shy away from the Sky News live debate that's scheduled for Tuesday evening, which he's been invited to [and] I've said I'm willing to go to.
"This is an audition to be prime minister of the United Kingdom, and Boris needs to show that he's prepared to answer difficult questions."
Mr Johnson appeared to deliver a retort to his rival to be prime minister as he promised "we are not going to bottle it" on the EU exit date of October 31st - deal or no-deal.
Mr Johnson remains under pressure to explain why police were called to the home he shares with partner Carrie Symonds, but in his column on Monday for the Daily Telegraph, he attempted to switch attention back to political rather than personal issues.
He wrote: "We must leave the EU on Oct 31 come what may. It will honour the referendum result, it will focus the minds of EU negotiators.
"It is absolutely vital that we keep our eyes on the prize. It has been a long and parching march - but the oasis is finally in sight.
"We are just over four months away from the date on which, by law, we must leave the EU; and this time we are not going to bottle it. We are not going to fail.
"This time we are not going to shrink in fear from the exit, as we have on the last two occasions."
Mr Johnson's campaign was rocked by revelations that police officers were called to the south London flat he shares with Ms Symonds by a neighbour who claimed to have been "frightened and concerned" after hearing shouting, "a loud scream" and banging coming from the property.
The police attended and found everyone was safe, with no offence having occurred.
When asked in Birmingham on Saturday about the incident, Mr Johnson said people "are entitled to ask about me and my determination, my character and what I want to do for the country", but he would not elaborate on the police visit.
Despite increasing calls to explain what happened, Mr Johnson has steadfastly refused to comment.
On Saturday, the neighbour who rang the police went public after suggestions that his recording of the row had been leaked to The Guardian with political intent.
Tom Penn said the allegations were "bizarre and fictitious", explaining in a statement to the paper that he dialled 999 after hearing shouting coming from his neighbour's flat.
The two leadership contenders are expected to take part in a head-to-head ITV debate on July 9th.
Sky News is also due to host a debate on Tuesday which Mr Hunt has agreed to take part in, but Mr Johnson has so far refused to commit.